LMFDB - Embedded newform 7800.2.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.10304.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 8 $ x^4 - 2x^3 - 7x^2 + 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1560)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16053$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +1.05545 q^{11} +1.00000 q^{13} +7.14949 q^{17} -6.61827 q^{19} +2.82843 q^{21} -3.49264 q^{23} -1.00000 q^{27} +4.82843 q^{29} -9.65685 q^{31} -1.05545 q^{33} +8.11091 q^{37} -1.00000 q^{39} -3.26561 q^{41} -7.44670 q^{43} +11.3765 q^{47} +1.00000 q^{49} -7.14949 q^{51} +4.53122 q^{53} +6.61827 q^{57} +1.05545 q^{59} -7.97792 q^{61} -2.82843 q^{63} +1.46878 q^{67} +3.49264 q^{69} +0.845296 q^{71} +9.28248 q^{73} -2.98527 q^{77} +6.85228 q^{79} +1.00000 q^{81} +7.97093 q^{83} -4.82843 q^{87} +0.139976 q^{89} -2.82843 q^{91} +9.65685 q^{93} -6.93933 q^{97} +1.05545 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 4 * q^9	$ 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^19 - 4 * q^23 - 4 * q^27 + 8 * q^29 - 16 * q^31 + 8 * q^33 + 8 * q^37 - 4 * q^39 - 4 * q^41 - 4 * q^43 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^57 - 8 * q^59 + 12 * q^61 + 24 * q^67 + 4 * q^69 - 12 * q^71 + 24 * q^73 + 8 * q^77 - 12 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^87 - 4 * q^89 + 16 * q^93 + 8 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.05545	0.318231	0.159115	−	0.987260i	$-0.449136\pi$
$11$	1.05545	0.318231	0.159115	+	0.987260i	$0.449136\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.14949	1.73401	0.867003	−	0.498303i	$-0.166043\pi$
$17$	7.14949	1.73401	0.867003	+	0.498303i	$0.166043\pi$
$18$	0	0
$19$	−6.61827	−1.51834	−0.759168	−	0.650895i	$-0.774394\pi$
$19$	−6.61827	−1.51834	−0.759168	+	0.650895i	$0.774394\pi$
$20$	0	0
$21$	2.82843	0.617213
$22$	0	0
$23$	−3.49264	−0.728265	−0.364132	−	0.931347i	$-0.618634\pi$
$23$	−3.49264	−0.728265	−0.364132	+	0.931347i	$0.618634\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	−9.65685	−1.73442	−0.867211	−	0.497941i	$-0.834090\pi$
$31$	−9.65685	−1.73442	−0.867211	+	0.497941i	$0.834090\pi$
$32$	0	0
$33$	−1.05545	−0.183731
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.11091	1.33342	0.666712	−	0.745315i	$-0.267701\pi$
$37$	8.11091	1.33342	0.666712	+	0.745315i	$0.267701\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.26561	−0.510003	−0.255001	−	0.966941i	$-0.582076\pi$
$41$	−3.26561	−0.510003	−0.255001	+	0.966941i	$0.582076\pi$
$42$	0	0
$43$	−7.44670	−1.13561	−0.567805	−	0.823163i	$-0.692207\pi$
$43$	−7.44670	−1.13561	−0.567805	+	0.823163i	$0.692207\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.3765	1.65944	0.829718	−	0.558183i	$-0.188501\pi$
$47$	11.3765	1.65944	0.829718	+	0.558183i	$0.188501\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.14949	−1.00113
$52$	0	0
$53$	4.53122	0.622411	0.311205	−	0.950343i	$-0.399267\pi$
$53$	4.53122	0.622411	0.311205	+	0.950343i	$0.399267\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.61827	0.876611
$58$	0	0
$59$	1.05545	0.137408	0.0687041	−	0.997637i	$-0.478114\pi$
$59$	1.05545	0.137408	0.0687041	+	0.997637i	$0.478114\pi$
$60$	0	0
$61$	−7.97792	−1.02147	−0.510734	−	0.859739i	$-0.670626\pi$
$61$	−7.97792	−1.02147	−0.510734	+	0.859739i	$0.670626\pi$
$62$	0	0
$63$	−2.82843	−0.356348
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.46878	0.179440	0.0897200	−	0.995967i	$-0.471403\pi$
$67$	1.46878	0.179440	0.0897200	+	0.995967i	$0.471403\pi$
$68$	0	0
$69$	3.49264	0.420464
$70$	0	0
$71$	0.845296	0.100318	0.0501591	−	0.998741i	$-0.484027\pi$
$71$	0.845296	0.100318	0.0501591	+	0.998741i	$0.484027\pi$
$72$	0	0
$73$	9.28248	1.08643	0.543216	−	0.839593i	$-0.317206\pi$
$73$	9.28248	1.08643	0.543216	+	0.839593i	$0.317206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.98527	−0.340203
$78$	0	0
$79$	6.85228	0.770942	0.385471	−	0.922720i	$-0.374039\pi$
$79$	6.85228	0.770942	0.385471	+	0.922720i	$0.374039\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.97093	0.874923	0.437462	−	0.899237i	$-0.355878\pi$
$83$	7.97093	0.874923	0.437462	+	0.899237i	$0.355878\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−4.82843	−0.517662
$88$	0	0
$89$	0.139976	0.0148374	0.00741870	−	0.999972i	$-0.497639\pi$
$89$	0.139976	0.0148374	0.00741870	+	0.999972i	$0.497639\pi$
$90$	0	0
$91$	−2.82843	−0.296500
$92$	0	0
$93$	9.65685	1.00137
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.93933	−0.704582	−0.352291	−	0.935890i	$-0.614597\pi$
$97$	−6.93933	−0.704582	−0.352291	+	0.935890i	$0.614597\pi$
$98$	0	0
$99$	1.05545	0.106077
$100$	0	0
$101$	−16.1421	−1.60620	−0.803101	−	0.595843i	$-0.796818\pi$
$101$	−16.1421	−1.60620	−0.803101	+	0.595843i	$0.796818\pi$
$102$	0	0
$103$	−16.7530	−1.65073	−0.825363	−	0.564603i	$-0.809029\pi$
$103$	−16.7530	−1.65073	−0.825363	+	0.564603i	$0.809029\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.87437	0.471223	0.235611	−	0.971847i	$-0.424291\pi$
$107$	4.87437	0.471223	0.235611	+	0.971847i	$0.424291\pi$
$108$	0	0
$109$	−7.14949	−0.684797	−0.342398	−	0.939555i	$-0.611239\pi$
$109$	−7.14949	−0.684797	−0.342398	+	0.939555i	$0.611239\pi$
$110$	0	0
$111$	−8.11091	−0.769853
$112$	0	0
$113$	−13.2604	−1.24743	−0.623717	−	0.781651i	$-0.714378\pi$
$113$	−13.2604	−1.24743	−0.623717	+	0.781651i	$0.714378\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−20.2218	−1.85373
$120$	0	0
$121$	−9.88602	−0.898729
$122$	0	0
$123$	3.26561	0.294450
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.3063	1.18075	0.590373	−	0.807130i	$-0.298981\pi$
$127$	13.3063	1.18075	0.590373	+	0.807130i	$0.298981\pi$
$128$	0	0
$129$	7.44670	0.655645
$130$	0	0
$131$	8.06497	0.704639	0.352320	−	0.935880i	$-0.385393\pi$
$131$	8.06497	0.704639	0.352320	+	0.935880i	$0.385393\pi$
$132$	0	0
$133$	18.7193	1.62317
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.8453	0.926576	0.463288	−	0.886208i	$-0.346670\pi$
$137$	10.8453	0.926576	0.463288	+	0.886208i	$0.346670\pi$
$138$	0	0
$139$	9.23654	0.783433	0.391717	−	0.920086i	$-0.371881\pi$
$139$	9.23654	0.783433	0.391717	+	0.920086i	$0.371881\pi$
$140$	0	0
$141$	−11.3765	−0.958075
$142$	0	0
$143$	1.05545	0.0882614
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−6.60140	−0.540808	−0.270404	−	0.962747i	$-0.587157\pi$
$149$	−6.60140	−0.540808	−0.270404	+	0.962747i	$0.587157\pi$
$150$	0	0
$151$	14.8934	1.21201	0.606004	−	0.795462i	$-0.292772\pi$
$151$	14.8934	1.21201	0.606004	+	0.795462i	$0.292772\pi$
$152$	0	0
$153$	7.14949	0.578002
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.3835	−0.908502	−0.454251	−	0.890874i	$-0.650093\pi$
$157$	−11.3835	−0.908502	−0.454251	+	0.890874i	$0.650093\pi$
$158$	0	0
$159$	−4.53122	−0.359349
$160$	0	0
$161$	9.87867	0.778548
$162$	0	0
$163$	−7.31371	−0.572854	−0.286427	−	0.958102i	$-0.592468\pi$
$163$	−7.31371	−0.572854	−0.286427	+	0.958102i	$0.592468\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.3428	1.18726	0.593630	−	0.804738i	$-0.297694\pi$
$167$	15.3428	1.18726	0.593630	+	0.804738i	$0.297694\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.61827	−0.506112
$172$	0	0
$173$	18.9706	1.44231	0.721153	−	0.692776i	$-0.243613\pi$
$173$	18.9706	1.44231	0.721153	+	0.692776i	$0.243613\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.05545	−0.0793327
$178$	0	0
$179$	−15.9246	−1.19026	−0.595130	−	0.803629i	$-0.702900\pi$
$179$	−15.9246	−1.19026	−0.595130	+	0.803629i	$0.702900\pi$
$180$	0	0
$181$	16.0772	1.19501	0.597503	−	0.801866i	$-0.296159\pi$
$181$	16.0772	1.19501	0.597503	+	0.801866i	$0.296159\pi$
$182$	0	0
$183$	7.97792	0.589745
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7.54595	0.551814
$188$	0	0
$189$	2.82843	0.205738
$190$	0	0
$191$	6.29898	0.455778	0.227889	−	0.973687i	$-0.426818\pi$
$191$	6.29898	0.455778	0.227889	+	0.973687i	$0.426818\pi$
$192$	0	0
$193$	−23.7695	−1.71097	−0.855484	−	0.517829i	$-0.826740\pi$
$193$	−23.7695	−1.71097	−0.855484	+	0.517829i	$0.826740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.62778	−0.543457	−0.271729	−	0.962374i	$-0.587595\pi$
$197$	−7.62778	−0.543457	−0.271729	+	0.962374i	$0.587595\pi$
$198$	0	0
$199$	6.85228	0.485745	0.242873	−	0.970058i	$-0.421910\pi$
$199$	6.85228	0.485745	0.242873	+	0.970058i	$0.421910\pi$
$200$	0	0
$201$	−1.46878	−0.103600
$202$	0	0
$203$	−13.6569	−0.958523
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−3.49264	−0.242755
$208$	0	0
$209$	−6.98527	−0.483181
$210$	0	0
$211$	15.8670	1.09233	0.546165	−	0.837678i	$-0.316087\pi$
$211$	15.8670	1.09233	0.546165	+	0.837678i	$0.316087\pi$
$212$	0	0
$213$	−0.845296	−0.0579187
$214$	0	0
$215$	0	0
$216$	0	0
$217$	27.3137	1.85418
$218$	0	0
$219$	−9.28248	−0.627252
$220$	0	0
$221$	7.14949	0.480927
$222$	0	0
$223$	−9.95406	−0.666573	−0.333287	−	0.942826i	$-0.608158\pi$
$223$	−9.95406	−0.666573	−0.333287	+	0.942826i	$0.608158\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.1547	0.740364	0.370182	−	0.928959i	$-0.379295\pi$
$227$	11.1547	0.740364	0.370182	+	0.928959i	$0.379295\pi$
$228$	0	0
$229$	23.9657	1.58370	0.791850	−	0.610716i	$-0.209118\pi$
$229$	23.9657	1.58370	0.791850	+	0.610716i	$0.209118\pi$
$230$	0	0
$231$	2.98527	0.196416
$232$	0	0
$233$	−15.3376	−1.00480	−0.502399	−	0.864636i	$-0.667549\pi$
$233$	−15.3376	−1.00480	−0.502399	+	0.864636i	$0.667549\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.85228	−0.445104
$238$	0	0
$239$	−14.7344	−0.953088	−0.476544	−	0.879150i	$-0.658111\pi$
$239$	−14.7344	−0.953088	−0.476544	+	0.879150i	$0.658111\pi$
$240$	0	0
$241$	10.1881	0.656272	0.328136	−	0.944631i	$-0.393580\pi$
$241$	10.1881	0.656272	0.328136	+	0.944631i	$0.393580\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.61827	−0.421110
$248$	0	0
$249$	−7.97093	−0.505137
$250$	0	0
$251$	15.8909	1.00302	0.501511	−	0.865151i	$-0.332778\pi$
$251$	15.8909	1.00302	0.501511	+	0.865151i	$0.332778\pi$
$252$	0	0
$253$	−3.68631	−0.231756
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.0239	1.24905	0.624527	−	0.781003i	$-0.285292\pi$
$257$	20.0239	1.24905	0.624527	+	0.781003i	$0.285292\pi$
$258$	0	0
$259$	−22.9411	−1.42549
$260$	0	0
$261$	4.82843	0.298872
$262$	0	0
$263$	27.0282	1.66663	0.833314	−	0.552800i	$-0.186441\pi$
$263$	27.0282	1.66663	0.833314	+	0.552800i	$0.186441\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−0.139976	−0.00856637
$268$	0	0
$269$	31.1274	1.89787	0.948936	−	0.315469i	$-0.102162\pi$
$269$	31.1274	1.89787	0.948936	+	0.315469i	$0.102162\pi$
$270$	0	0
$271$	21.0043	1.27592	0.637960	−	0.770069i	$-0.279778\pi$
$271$	21.0043	1.27592	0.637960	+	0.770069i	$0.279778\pi$
$272$	0	0
$273$	2.82843	0.171184
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.2145	0.793980	0.396990	−	0.917823i	$-0.370055\pi$
$277$	13.2145	0.793980	0.396990	+	0.917823i	$0.370055\pi$
$278$	0	0
$279$	−9.65685	−0.578141
$280$	0	0
$281$	1.83057	0.109202	0.0546012	−	0.998508i	$-0.482611\pi$
$281$	1.83057	0.109202	0.0546012	+	0.998508i	$0.482611\pi$
$282$	0	0
$283$	27.5355	1.63682	0.818408	−	0.574637i	$-0.194857\pi$
$283$	27.5355	1.63682	0.818408	+	0.574637i	$0.194857\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.23654	0.545216
$288$	0	0
$289$	34.1152	2.00678
$290$	0	0
$291$	6.93933	0.406791
$292$	0	0
$293$	3.95190	0.230873	0.115436	−	0.993315i	$-0.463173\pi$
$293$	3.95190	0.230873	0.115436	+	0.993315i	$0.463173\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.05545	−0.0612436
$298$	0	0
$299$	−3.49264	−0.201984
$300$	0	0
$301$	21.0624	1.21402
$302$	0	0
$303$	16.1421	0.927341
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−14.9271	−0.851936	−0.425968	−	0.904738i	$-0.640066\pi$
$307$	−14.9271	−0.851936	−0.425968	+	0.904738i	$0.640066\pi$
$308$	0	0
$309$	16.7530	0.953047
$310$	0	0
$311$	4.87437	0.276400	0.138200	−	0.990404i	$-0.455868\pi$
$311$	4.87437	0.276400	0.138200	+	0.990404i	$0.455868\pi$
$312$	0	0
$313$	27.2145	1.53825	0.769126	−	0.639097i	$-0.220692\pi$
$313$	27.2145	1.53825	0.769126	+	0.639097i	$0.220692\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.68592	0.207022	0.103511	−	0.994628i	$-0.466992\pi$
$317$	3.68592	0.207022	0.103511	+	0.994628i	$0.466992\pi$
$318$	0	0
$319$	5.09618	0.285331
$320$	0	0
$321$	−4.87437	−0.272061
$322$	0	0
$323$	−47.3173	−2.63280
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.14949	0.395368
$328$	0	0
$329$	−32.1776	−1.77401
$330$	0	0
$331$	−24.6850	−1.35681	−0.678405	−	0.734688i	$-0.737329\pi$
$331$	−24.6850	−1.35681	−0.678405	+	0.734688i	$0.737329\pi$
$332$	0	0
$333$	8.11091	0.444475
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.5282	−1.66298	−0.831488	−	0.555543i	$-0.812510\pi$
$337$	−30.5282	−1.66298	−0.831488	+	0.555543i	$0.812510\pi$
$338$	0	0
$339$	13.2604	0.720206
$340$	0	0
$341$	−10.1924	−0.551947
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.32844	−0.447094	−0.223547	−	0.974693i	$-0.571764\pi$
$347$	−8.32844	−0.447094	−0.223547	+	0.974693i	$0.571764\pi$
$348$	0	0
$349$	−8.57410	−0.458961	−0.229481	−	0.973313i	$-0.573703\pi$
$349$	−8.57410	−0.458961	−0.229481	+	0.973313i	$0.573703\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	2.42498	0.129069	0.0645344	−	0.997915i	$-0.479444\pi$
$353$	2.42498	0.129069	0.0645344	+	0.997915i	$0.479444\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	20.2218	1.07025
$358$	0	0
$359$	−18.5499	−0.979024	−0.489512	−	0.871997i	$-0.662825\pi$
$359$	−18.5499	−0.979024	−0.489512	+	0.871997i	$0.662825\pi$
$360$	0	0
$361$	24.8015	1.30534
$362$	0	0
$363$	9.88602	0.518881
$364$	0	0
$365$	0	0
$366$	0	0
$367$	5.29973	0.276644	0.138322	−	0.990387i	$-0.455829\pi$
$367$	5.29973	0.276644	0.138322	+	0.990387i	$0.455829\pi$
$368$	0	0
$369$	−3.26561	−0.170001
$370$	0	0
$371$	−12.8162	−0.665385
$372$	0	0
$373$	29.2071	1.51229	0.756143	−	0.654407i	$-0.227082\pi$
$373$	29.2071	1.51229	0.756143	+	0.654407i	$0.227082\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.82843	0.248677
$378$	0	0
$379$	10.3523	0.531762	0.265881	−	0.964006i	$-0.414337\pi$
$379$	10.3523	0.531762	0.265881	+	0.964006i	$0.414337\pi$
$380$	0	0
$381$	−13.3063	−0.681704
$382$	0	0
$383$	30.0818	1.53711	0.768555	−	0.639784i	$-0.220976\pi$
$383$	30.0818	1.53711	0.768555	+	0.639784i	$0.220976\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−7.44670	−0.378537
$388$	0	0
$389$	33.2383	1.68525	0.842625	−	0.538501i	$-0.181009\pi$
$389$	33.2383	1.68525	0.842625	+	0.538501i	$0.181009\pi$
$390$	0	0
$391$	−24.9706	−1.26282
$392$	0	0
$393$	−8.06497	−0.406824
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.5018	−0.677635	−0.338818	−	0.940852i	$-0.610027\pi$
$397$	−13.5018	−0.677635	−0.338818	+	0.940852i	$0.610027\pi$
$398$	0	0
$399$	−18.7193	−0.937137
$400$	0	0
$401$	3.53159	0.176359	0.0881795	−	0.996105i	$-0.471895\pi$
$401$	3.53159	0.176359	0.0881795	+	0.996105i	$0.471895\pi$
$402$	0	0
$403$	−9.65685	−0.481042
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.56068	0.424337
$408$	0	0
$409$	−24.7530	−1.22396	−0.611979	−	0.790874i	$-0.709626\pi$
$409$	−24.7530	−1.22396	−0.611979	+	0.790874i	$0.709626\pi$
$410$	0	0
$411$	−10.8453	−0.534959
$412$	0	0
$413$	−2.98527	−0.146896
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−9.23654	−0.452315
$418$	0	0
$419$	5.39338	0.263484	0.131742	−	0.991284i	$-0.457943\pi$
$419$	5.39338	0.263484	0.131742	+	0.991284i	$0.457943\pi$
$420$	0	0
$421$	32.4337	1.58072	0.790362	−	0.612640i	$-0.209893\pi$
$421$	32.4337	1.58072	0.790362	+	0.612640i	$0.209893\pi$
$422$	0	0
$423$	11.3765	0.553145
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.5650	1.09199
$428$	0	0
$429$	−1.05545	−0.0509577
$430$	0	0
$431$	−31.5983	−1.52204	−0.761019	−	0.648730i	$-0.775301\pi$
$431$	−31.5983	−1.52204	−0.761019	+	0.648730i	$0.775301\pi$
$432$	0	0
$433$	27.8566	1.33870	0.669351	−	0.742946i	$-0.266572\pi$
$433$	27.8566	1.33870	0.669351	+	0.742946i	$0.266572\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	23.1152	1.10575
$438$	0	0
$439$	7.09190	0.338478	0.169239	−	0.985575i	$-0.445869\pi$
$439$	7.09190	0.338478	0.169239	+	0.985575i	$0.445869\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	0.386578	0.0183669	0.00918343	−	0.999958i	$-0.497077\pi$
$443$	0.386578	0.0183669	0.00918343	+	0.999958i	$0.497077\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	6.60140	0.312236
$448$	0	0
$449$	1.18844	0.0560860	0.0280430	−	0.999607i	$-0.491072\pi$
$449$	1.18844	0.0560860	0.0280430	+	0.999607i	$0.491072\pi$
$450$	0	0
$451$	−3.44670	−0.162299
$452$	0	0
$453$	−14.8934	−0.699753
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.5434	0.493200	0.246600	−	0.969117i	$-0.420687\pi$
$457$	10.5434	0.493200	0.246600	+	0.969117i	$0.420687\pi$
$458$	0	0
$459$	−7.14949	−0.333710
$460$	0	0
$461$	11.8961	0.554056	0.277028	−	0.960862i	$-0.410651\pi$
$461$	11.8961	0.554056	0.277028	+	0.960862i	$0.410651\pi$
$462$	0	0
$463$	1.92032	0.0892451	0.0446225	−	0.999004i	$-0.485791\pi$
$463$	1.92032	0.0892451	0.0446225	+	0.999004i	$0.485791\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1256	0.885029	0.442514	−	0.896761i	$-0.354087\pi$
$467$	19.1256	0.885029	0.442514	+	0.896761i	$0.354087\pi$
$468$	0	0
$469$	−4.15434	−0.191829
$470$	0	0
$471$	11.3835	0.524524
$472$	0	0
$473$	−7.85964	−0.361386
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.53122	0.207470
$478$	0	0
$479$	12.5793	0.574764	0.287382	−	0.957816i	$-0.407215\pi$
$479$	12.5793	0.574764	0.287382	+	0.957816i	$0.407215\pi$
$480$	0	0
$481$	8.11091	0.369825
$482$	0	0
$483$	−9.87867	−0.449495
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.8180	1.03398	0.516991	−	0.855991i	$-0.327052\pi$
$487$	22.8180	1.03398	0.516991	+	0.855991i	$0.327052\pi$
$488$	0	0
$489$	7.31371	0.330737
$490$	0	0
$491$	38.6924	1.74616	0.873081	−	0.487574i	$-0.162118\pi$
$491$	38.6924	1.74616	0.873081	+	0.487574i	$0.162118\pi$
$492$	0	0
$493$	34.5208	1.55474
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.39086	−0.107245
$498$	0	0
$499$	−26.5404	−1.18811	−0.594055	−	0.804424i	$-0.702474\pi$
$499$	−26.5404	−1.18811	−0.594055	+	0.804424i	$0.702474\pi$
$500$	0	0
$501$	−15.3428	−0.685465
$502$	0	0
$503$	−15.4008	−0.686686	−0.343343	−	0.939210i	$-0.611559\pi$
$503$	−15.4008	−0.686686	−0.343343	+	0.939210i	$0.611559\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−30.1846	−1.33791	−0.668955	−	0.743303i	$-0.733258\pi$
$509$	−30.1846	−1.33791	−0.668955	+	0.743303i	$0.733258\pi$
$510$	0	0
$511$	−26.2548	−1.16144
$512$	0	0
$513$	6.61827	0.292204
$514$	0	0
$515$	0	0
$516$	0	0
$517$	12.0074	0.528084
$518$	0	0
$519$	−18.9706	−0.832715
$520$	0	0
$521$	−26.7825	−1.17336	−0.586681	−	0.809818i	$-0.699566\pi$
$521$	−26.7825	−1.17336	−0.586681	+	0.809818i	$0.699566\pi$
$522$	0	0
$523$	27.8670	1.21854	0.609270	−	0.792963i	$-0.291463\pi$
$523$	27.8670	1.21854	0.609270	+	0.792963i	$0.291463\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−69.0416	−3.00750
$528$	0	0
$529$	−10.8015	−0.469630
$530$	0	0
$531$	1.05545	0.0458027
$532$	0	0
$533$	−3.26561	−0.141449
$534$	0	0
$535$	0	0
$536$	0	0
$537$	15.9246	0.687197
$538$	0	0
$539$	1.05545	0.0454616
$540$	0	0
$541$	20.8982	0.898485	0.449242	−	0.893410i	$-0.351694\pi$
$541$	20.8982	0.898485	0.449242	+	0.893410i	$0.351694\pi$
$542$	0	0
$543$	−16.0772	−0.689937
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24.3106	1.03945	0.519724	−	0.854334i	$-0.326035\pi$
$547$	24.3106	1.03945	0.519724	+	0.854334i	$0.326035\pi$
$548$	0	0
$549$	−7.97792	−0.340489
$550$	0	0
$551$	−31.9558	−1.36136
$552$	0	0
$553$	−19.3812	−0.824172
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.7197	−0.454207	−0.227104	−	0.973871i	$-0.572926\pi$
$557$	−10.7197	−0.454207	−0.227104	+	0.973871i	$0.572926\pi$
$558$	0	0
$559$	−7.44670	−0.314962
$560$	0	0
$561$	−7.54595	−0.318590
$562$	0	0
$563$	24.6898	1.04055	0.520276	−	0.853998i	$-0.325829\pi$
$563$	24.6898	1.04055	0.520276	+	0.853998i	$0.325829\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.82843	−0.118783
$568$	0	0
$569$	−35.6023	−1.49252	−0.746262	−	0.665652i	$-0.768153\pi$
$569$	−35.6023	−1.49252	−0.746262	+	0.665652i	$0.768153\pi$
$570$	0	0
$571$	−13.3254	−0.557649	−0.278825	−	0.960342i	$-0.589945\pi$
$571$	−13.3254	−0.557649	−0.278825	+	0.960342i	$0.589945\pi$
$572$	0	0
$573$	−6.29898	−0.263144
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.3112	1.09535	0.547674	−	0.836692i	$-0.315513\pi$
$577$	26.3112	1.09535	0.547674	+	0.836692i	$0.315513\pi$
$578$	0	0
$579$	23.7695	0.987828
$580$	0	0
$581$	−22.5452	−0.935332
$582$	0	0
$583$	4.78249	0.198070
$584$	0	0
$585$	0	0
$586$	0	0
$587$	28.3808	1.17140	0.585701	−	0.810527i	$-0.300819\pi$
$587$	28.3808	1.17140	0.585701	+	0.810527i	$0.300819\pi$
$588$	0	0
$589$	63.9117	2.63343
$590$	0	0
$591$	7.62778	0.313765
$592$	0	0
$593$	−6.81156	−0.279717	−0.139859	−	0.990171i	$-0.544665\pi$
$593$	−6.81156	−0.279717	−0.139859	+	0.990171i	$0.544665\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.85228	−0.280445
$598$	0	0
$599$	38.9307	1.59066	0.795332	−	0.606174i	$-0.207296\pi$
$599$	38.9307	1.59066	0.795332	+	0.606174i	$0.207296\pi$
$600$	0	0
$601$	−2.87131	−0.117123	−0.0585616	−	0.998284i	$-0.518651\pi$
$601$	−2.87131	−0.117123	−0.0585616	+	0.998284i	$0.518651\pi$
$602$	0	0
$603$	1.46878	0.0598133
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−10.9221	−0.443313	−0.221657	−	0.975125i	$-0.571146\pi$
$607$	−10.9221	−0.443313	−0.221657	+	0.975125i	$0.571146\pi$
$608$	0	0
$609$	13.6569	0.553404
$610$	0	0
$611$	11.3765	0.460245
$612$	0	0
$613$	−38.8779	−1.57026	−0.785132	−	0.619328i	$-0.787405\pi$
$613$	−38.8779	−1.57026	−0.785132	+	0.619328i	$0.787405\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.09656	−0.0441460	−0.0220730	−	0.999756i	$-0.507027\pi$
$617$	−1.09656	−0.0441460	−0.0220730	+	0.999756i	$0.507027\pi$
$618$	0	0
$619$	−5.52209	−0.221952	−0.110976	−	0.993823i	$-0.535398\pi$
$619$	−5.52209	−0.221952	−0.110976	+	0.993823i	$0.535398\pi$
$620$	0	0
$621$	3.49264	0.140155
$622$	0	0
$623$	−0.395911	−0.0158618
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.98527	0.278965
$628$	0	0
$629$	57.9888	2.31217
$630$	0	0
$631$	−1.74873	−0.0696159	−0.0348079	−	0.999394i	$-0.511082\pi$
$631$	−1.74873	−0.0696159	−0.0348079	+	0.999394i	$0.511082\pi$
$632$	0	0
$633$	−15.8670	−0.630657
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	0.845296	0.0334394
$640$	0	0
$641$	2.73400	0.107987	0.0539933	−	0.998541i	$-0.482805\pi$
$641$	2.73400	0.107987	0.0539933	+	0.998541i	$0.482805\pi$
$642$	0	0
$643$	−36.3180	−1.43224	−0.716121	−	0.697976i	$-0.754084\pi$
$643$	−36.3180	−1.43224	−0.716121	+	0.697976i	$0.754084\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.4295	1.11768	0.558839	−	0.829276i	$-0.311247\pi$
$647$	28.4295	1.11768	0.558839	+	0.829276i	$0.311247\pi$
$648$	0	0
$649$	1.11398	0.0437276
$650$	0	0
$651$	−27.3137	−1.07051
$652$	0	0
$653$	7.48275	0.292823	0.146411	−	0.989224i	$-0.453228\pi$
$653$	7.48275	0.292823	0.146411	+	0.989224i	$0.453228\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	9.28248	0.362144
$658$	0	0
$659$	−1.90635	−0.0742609	−0.0371304	−	0.999310i	$-0.511822\pi$
$659$	−1.90635	−0.0742609	−0.0371304	+	0.999310i	$0.511822\pi$
$660$	0	0
$661$	23.9761	0.932564	0.466282	−	0.884636i	$-0.345593\pi$
$661$	23.9761	0.932564	0.466282	+	0.884636i	$0.345593\pi$
$662$	0	0
$663$	−7.14949	−0.277663
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−16.8639	−0.652974
$668$	0	0
$669$	9.95406	0.384846
$670$	0	0
$671$	−8.42031	−0.325063
$672$	0	0
$673$	−18.6127	−0.717466	−0.358733	−	0.933440i	$-0.616791\pi$
$673$	−18.6127	−0.717466	−0.358733	+	0.933440i	$0.616791\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32.7530	−1.25880	−0.629401	−	0.777081i	$-0.716700\pi$
$677$	−32.7530	−1.25880	−0.629401	+	0.777081i	$0.716700\pi$
$678$	0	0
$679$	19.6274	0.753230
$680$	0	0
$681$	−11.1547	−0.427449
$682$	0	0
$683$	34.5976	1.32384	0.661920	−	0.749575i	$-0.269742\pi$
$683$	34.5976	1.32384	0.661920	+	0.749575i	$0.269742\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−23.9657	−0.914349
$688$	0	0
$689$	4.53122	0.172626
$690$	0	0
$691$	3.99517	0.151984	0.0759918	−	0.997108i	$-0.475788\pi$
$691$	3.99517	0.151984	0.0759918	+	0.997108i	$0.475788\pi$
$692$	0	0
$693$	−2.98527	−0.113401
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.3474	−0.884348
$698$	0	0
$699$	15.3376	0.580120
$700$	0	0
$701$	−0.267750	−0.0101128	−0.00505638	−	0.999987i	$-0.501610\pi$
$701$	−0.267750	−0.0101128	−0.00505638	+	0.999987i	$0.501610\pi$
$702$	0	0
$703$	−53.6802	−2.02459
$704$	0	0
$705$	0	0
$706$	0	0
$707$	45.6569	1.71710
$708$	0	0
$709$	−1.13478	−0.0426176	−0.0213088	−	0.999773i	$-0.506783\pi$
$709$	−1.13478	−0.0426176	−0.0213088	+	0.999773i	$0.506783\pi$
$710$	0	0
$711$	6.85228	0.256981
$712$	0	0
$713$	33.7279	1.26312
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.7344	0.550266
$718$	0	0
$719$	−14.0434	−0.523732	−0.261866	−	0.965104i	$-0.584338\pi$
$719$	−14.0434	−0.523732	−0.261866	+	0.965104i	$0.584338\pi$
$720$	0	0
$721$	47.3847	1.76470
$722$	0	0
$723$	−10.1881	−0.378899
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.5099	−0.871934	−0.435967	−	0.899963i	$-0.643593\pi$
$727$	−23.5099	−0.871934	−0.435967	+	0.899963i	$0.643593\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−53.2401	−1.96916
$732$	0	0
$733$	−21.6274	−0.798826	−0.399413	−	0.916771i	$-0.630786\pi$
$733$	−21.6274	−0.798826	−0.399413	+	0.916771i	$0.630786\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.55023	0.0571034
$738$	0	0
$739$	−26.8738	−0.988569	−0.494285	−	0.869300i	$-0.664570\pi$
$739$	−26.8738	−0.988569	−0.494285	+	0.869300i	$0.664570\pi$
$740$	0	0
$741$	6.61827	0.243128
$742$	0	0
$743$	11.5506	0.423751	0.211875	−	0.977297i	$-0.432043\pi$
$743$	11.5506	0.423751	0.211875	+	0.977297i	$0.432043\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	7.97093	0.291641
$748$	0	0
$749$	−13.7868	−0.503758
$750$	0	0
$751$	52.5980	1.91933	0.959663	−	0.281151i	$-0.0907163\pi$
$751$	52.5980	1.91933	0.959663	+	0.281151i	$0.0907163\pi$
$752$	0	0
$753$	−15.8909	−0.579095
$754$	0	0
$755$	0	0
$756$	0	0
$757$	18.8406	0.684774	0.342387	−	0.939559i	$-0.388765\pi$
$757$	18.8406	0.684774	0.342387	+	0.939559i	$0.388765\pi$
$758$	0	0
$759$	3.68631	0.133805
$760$	0	0
$761$	−49.3323	−1.78830	−0.894148	−	0.447771i	$-0.852218\pi$
$761$	−49.3323	−1.78830	−0.894148	+	0.447771i	$0.852218\pi$
$762$	0	0
$763$	20.2218	0.732079
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.05545	0.0381102
$768$	0	0
$769$	32.6274	1.17657	0.588287	−	0.808652i	$-0.299802\pi$
$769$	32.6274	1.17657	0.588287	+	0.808652i	$0.299802\pi$
$770$	0	0
$771$	−20.0239	−0.721142
$772$	0	0
$773$	−19.0229	−0.684208	−0.342104	−	0.939662i	$-0.611139\pi$
$773$	−19.0229	−0.684208	−0.342104	+	0.939662i	$0.611139\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	22.9411	0.823008
$778$	0	0
$779$	21.6127	0.774355
$780$	0	0
$781$	0.892170	0.0319243
$782$	0	0
$783$	−4.82843	−0.172554
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−3.79216	−0.135176	−0.0675880	−	0.997713i	$-0.521530\pi$
$787$	−3.79216	−0.135176	−0.0675880	+	0.997713i	$0.521530\pi$
$788$	0	0
$789$	−27.0282	−0.962228
$790$	0	0
$791$	37.5061	1.33356
$792$	0	0
$793$	−7.97792	−0.283304
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.0287	−1.09909	−0.549547	−	0.835463i	$-0.685199\pi$
$797$	−31.0287	−1.09909	−0.549547	+	0.835463i	$0.685199\pi$
$798$	0	0
$799$	81.3363	2.87747
$800$	0	0
$801$	0.139976	0.00494580
$802$	0	0
$803$	9.79722	0.345736
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−31.1274	−1.09574
$808$	0	0
$809$	49.2211	1.73052	0.865260	−	0.501323i	$-0.167153\pi$
$809$	49.2211	1.73052	0.865260	+	0.501323i	$0.167153\pi$
$810$	0	0
$811$	8.76291	0.307707	0.153854	−	0.988094i	$-0.450832\pi$
$811$	8.76291	0.307707	0.153854	+	0.988094i	$0.450832\pi$
$812$	0	0
$813$	−21.0043	−0.736653
$814$	0	0
$815$	0	0
$816$	0	0
$817$	49.2843	1.72424
$818$	0	0
$819$	−2.82843	−0.0988332
$820$	0	0
$821$	24.3060	0.848284	0.424142	−	0.905596i	$-0.360576\pi$
$821$	24.3060	0.848284	0.424142	+	0.905596i	$0.360576\pi$
$822$	0	0
$823$	43.1586	1.50442	0.752208	−	0.658926i	$-0.228989\pi$
$823$	43.1586	1.50442	0.752208	+	0.658926i	$0.228989\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−27.7189	−0.963882	−0.481941	−	0.876204i	$-0.660068\pi$
$827$	−27.7189	−0.963882	−0.481941	+	0.876204i	$0.660068\pi$
$828$	0	0
$829$	23.2094	0.806096	0.403048	−	0.915179i	$-0.367951\pi$
$829$	23.2094	0.806096	0.403048	+	0.915179i	$0.367951\pi$
$830$	0	0
$831$	−13.2145	−0.458404
$832$	0	0
$833$	7.14949	0.247715
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.65685	0.333790
$838$	0	0
$839$	−25.2796	−0.872748	−0.436374	−	0.899765i	$-0.643738\pi$
$839$	−25.2796	−0.872748	−0.436374	+	0.899765i	$0.643738\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	0	0
$843$	−1.83057	−0.0630481
$844$	0	0
$845$	0	0
$846$	0	0
$847$	27.9619	0.960782
$848$	0	0
$849$	−27.5355	−0.945017
$850$	0	0
$851$	−28.3284	−0.971086
$852$	0	0
$853$	4.43934	0.152000	0.0760001	−	0.997108i	$-0.475785\pi$
$853$	4.43934	0.152000	0.0760001	+	0.997108i	$0.475785\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	31.3609	1.07127	0.535633	−	0.844451i	$-0.320073\pi$
$857$	31.3609	1.07127	0.535633	+	0.844451i	$0.320073\pi$
$858$	0	0
$859$	−28.7457	−0.980791	−0.490395	−	0.871500i	$-0.663148\pi$
$859$	−28.7457	−0.980791	−0.490395	+	0.871500i	$0.663148\pi$
$860$	0	0
$861$	−9.23654	−0.314780
$862$	0	0
$863$	36.3471	1.23727	0.618634	−	0.785679i	$-0.287686\pi$
$863$	36.3471	1.23727	0.618634	+	0.785679i	$0.287686\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−34.1152	−1.15861
$868$	0	0
$869$	7.23226	0.245338
$870$	0	0
$871$	1.46878	0.0497677
$872$	0	0
$873$	−6.93933	−0.234861
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21.9896	−0.742535	−0.371268	−	0.928526i	$-0.621077\pi$
$877$	−21.9896	−0.742535	−0.371268	+	0.928526i	$0.621077\pi$
$878$	0	0
$879$	−3.95190	−0.133294
$880$	0	0
$881$	−44.5348	−1.50041	−0.750207	−	0.661203i	$-0.770046\pi$
$881$	−44.5348	−1.50041	−0.750207	+	0.661203i	$0.770046\pi$
$882$	0	0
$883$	6.05277	0.203692	0.101846	−	0.994800i	$-0.467525\pi$
$883$	6.05277	0.203692	0.101846	+	0.994800i	$0.467525\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.9700	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$887$	35.9700	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$888$	0	0
$889$	−37.6360	−1.26227
$890$	0	0
$891$	1.05545	0.0353590
$892$	0	0
$893$	−75.2929	−2.51958
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.49264	0.116616
$898$	0	0
$899$	−46.6274	−1.55511
$900$	0	0
$901$	32.3959	1.07926
$902$	0	0
$903$	−21.0624	−0.700914
$904$	0	0
$905$	0	0
$906$	0	0
$907$	39.2695	1.30392	0.651962	−	0.758252i	$-0.273946\pi$
$907$	39.2695	1.30392	0.651962	+	0.758252i	$0.273946\pi$
$908$	0	0
$909$	−16.1421	−0.535401
$910$	0	0
$911$	−26.4583	−0.876604	−0.438302	−	0.898828i	$-0.644420\pi$
$911$	−26.4583	−0.876604	−0.438302	+	0.898828i	$0.644420\pi$
$912$	0	0
$913$	8.41294	0.278428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−22.8112	−0.753291
$918$	0	0
$919$	−37.1513	−1.22551	−0.612754	−	0.790274i	$-0.709938\pi$
$919$	−37.1513	−1.22551	−0.612754	+	0.790274i	$0.709938\pi$
$920$	0	0
$921$	14.9271	0.491866
$922$	0	0
$923$	0.845296	0.0278232
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.7530	−0.550242
$928$	0	0
$929$	15.2412	0.500048	0.250024	−	0.968240i	$-0.419562\pi$
$929$	15.2412	0.500048	0.250024	+	0.968240i	$0.419562\pi$
$930$	0	0
$931$	−6.61827	−0.216905
$932$	0	0
$933$	−4.87437	−0.159580
$934$	0	0
$935$	0	0
$936$	0	0
$937$	28.1850	0.920764	0.460382	−	0.887721i	$-0.347712\pi$
$937$	28.1850	0.920764	0.460382	+	0.887721i	$0.347712\pi$
$938$	0	0
$939$	−27.2145	−0.888110
$940$	0	0
$941$	42.8752	1.39769	0.698846	−	0.715272i	$-0.253697\pi$
$941$	42.8752	1.39769	0.698846	+	0.715272i	$0.253697\pi$
$942$	0	0
$943$	11.4056	0.371417
$944$	0	0
$945$	0	0
$946$	0	0
$947$	11.3428	0.368591	0.184295	−	0.982871i	$-0.441000\pi$
$947$	11.3428	0.368591	0.184295	+	0.982871i	$0.441000\pi$
$948$	0	0
$949$	9.28248	0.301322
$950$	0	0
$951$	−3.68592	−0.119524
$952$	0	0
$953$	−0.754313	−0.0244346	−0.0122173	−	0.999925i	$-0.503889\pi$
$953$	−0.754313	−0.0244346	−0.0122173	+	0.999925i	$0.503889\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−5.09618	−0.164736
$958$	0	0
$959$	−30.6751	−0.990552
$960$	0	0
$961$	62.2548	2.00822
$962$	0	0
$963$	4.87437	0.157074
$964$	0	0
$965$	0	0
$966$	0	0
$967$	17.2530	0.554820	0.277410	−	0.960752i	$-0.410524\pi$
$967$	17.2530	0.554820	0.277410	+	0.960752i	$0.410524\pi$
$968$	0	0
$969$	47.3173	1.52005
$970$	0	0
$971$	−35.2825	−1.13227	−0.566134	−	0.824313i	$-0.691562\pi$
$971$	−35.2825	−1.13227	−0.566134	+	0.824313i	$0.691562\pi$
$972$	0	0
$973$	−26.1249	−0.837525
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9.04808	−0.289474	−0.144737	−	0.989470i	$-0.546234\pi$
$977$	−9.04808	−0.289474	−0.144737	+	0.989470i	$0.546234\pi$
$978$	0	0
$979$	0.147738	0.00472172
$980$	0	0
$981$	−7.14949	−0.228266
$982$	0	0
$983$	−30.5255	−0.973611	−0.486805	−	0.873510i	$-0.661838\pi$
$983$	−30.5255	−0.973611	−0.486805	+	0.873510i	$0.661838\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	32.1776	1.02423
$988$	0	0
$989$	26.0086	0.827025
$990$	0	0
$991$	−6.10048	−0.193788	−0.0968940	−	0.995295i	$-0.530891\pi$
$991$	−6.10048	−0.193788	−0.0968940	+	0.995295i	$0.530891\pi$
$992$	0	0
$993$	24.6850	0.783355
$994$	0	0
$995$	0	0
$996$	0	0
$997$	31.4510	0.996062	0.498031	−	0.867159i	$-0.334056\pi$
$997$	31.4510	0.996062	0.498031	+	0.867159i	$0.334056\pi$
$998$	0	0
$999$	−8.11091	−0.256618

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bv.1.2		4
5.2	odd	4		1560.2.l.e.1249.5	yes	8
5.3	odd	4		1560.2.l.e.1249.1	✓	8
5.4	even	2		7800.2.a.bw.1.4		4
15.2	even	4		4680.2.l.f.2809.7		8
15.8	even	4		4680.2.l.f.2809.8		8
20.3	even	4		3120.2.l.o.1249.5		8
20.7	even	4		3120.2.l.o.1249.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.2.l.e.1249.1	✓	8	5.3	odd	4
1560.2.l.e.1249.5	yes	8	5.2	odd	4
3120.2.l.o.1249.1		8	20.7	even	4
3120.2.l.o.1249.5		8	20.3	even	4
4680.2.l.f.2809.7		8	15.2	even	4
4680.2.l.f.2809.8		8	15.8	even	4
7800.2.a.bv.1.2		4	1.1	even	1	trivial
7800.2.a.bw.1.4		4	5.4	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.82843 q^{7} +1.00000 q^{9} +1.05545 q^{11} +1.00000 q^{13} +7.14949 q^{17} -6.61827 q^{19} +2.82843 q^{21} -3.49264 q^{23} -1.00000 q^{27} +4.82843 q^{29} -9.65685 q^{31} -1.05545 q^{33} +8.11091 q^{37} -1.00000 q^{39} -3.26561 q^{41} -7.44670 q^{43} +11.3765 q^{47} +1.00000 q^{49} -7.14949 q^{51} +4.53122 q^{53} +6.61827 q^{57} +1.05545 q^{59} -7.97792 q^{61} -2.82843 q^{63} +1.46878 q^{67} +3.49264 q^{69} +0.845296 q^{71} +9.28248 q^{73} -2.98527 q^{77} +6.85228 q^{79} +1.00000 q^{81} +7.97093 q^{83} -4.82843 q^{87} +0.139976 q^{89} -2.82843 q^{91} +9.65685 q^{93} -6.93933 q^{97} +1.05545 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 4 * q^9	\( 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 4 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^17 - 12 * q^19 - 4 * q^23 - 4 * q^27 + 8 * q^29 - 16 * q^31 + 8 * q^33 + 8 * q^37 - 4 * q^39 - 4 * q^41 - 4 * q^43 + 12 * q^47 + 4 * q^49 + 4 * q^51 + 12 * q^57 - 8 * q^59 + 12 * q^61 + 24 * q^67 + 4 * q^69 - 12 * q^71 + 24 * q^73 + 8 * q^77 - 12 * q^79 + 4 * q^81 + 12 * q^83 - 8 * q^87 - 4 * q^89 + 16 * q^93 + 8 * q^97 - 8 * q^99

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